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The real Seifert form and the spectral pairs of isolated hypersurface singularities. (English) Zbl 0851.14015
For a hypersurface singularity \(f: (\mathbb{C}^{n+1}, 0)\to (\mathbb{C}, 0)\), a discrete invariant \(\text{Spp} (f)\) called the set of spectral pairs is defined as an element in the free abelian group generated by \(\mathbb{Q} \times \mathbb{N}\). It determines and is determined by the collection of Hodge numbers of the cohomology of the Milnor fiber of \(f\). The main result of this article stated in the beginning part is the following:
Theorem. Consider the image \(\text{Spp}_{\text{mod-}2} (f)\) of \(\text{Spp} (f)\) under the projection induced by \[ \mathbb{Q} \times \mathbb{N}\to (\mathbb{Q}/ 2\mathbb{Z}) \times \mathbb{N}. \] Then the information contained in \(\text{Spp}_{\text{mod-}2} (f)\) is equivalent to the information contained in the real Seifert form of the singularity \(f\).
Needless to say, this theorem has no meaning unless the meaning of the phrase “the information contained in the real Seifert form of the singularity \(f\)” is made clear. Let \(F\) denote the Milnor fiber of \(f\) and let \(U= \widetilde {H}_n (F, \mathbb{R})\). The monodromy defines an isomorphism \(h: U\to U\) and the intersection form defines a map \(b:U\to U^*= \operatorname{Hom} (U, \mathbb{R})\). Besides, in case \(n>0\) we have the canonical identification \(U^*= H_n (F, \partial F, \mathbb{R})\) and the variation isomorphism \(\text{Var}: U^*\to U\) is defined. The real Seifert form \(L\) is the bilinear form on \(U\) defined by \(L(a, b)= \langle \text{Var}^{-1} (a), b\rangle\) for \(a, b\in U\), where \(\langle\;, \;\rangle: U^* \times U\to \mathbb{R}\) denotes the canonical pairing. The author defines two bilinear forms on real vector spaces \(U_1\) and \(U_2\) to be isomorphic, if their induced hermitian forms on \(U_1 \otimes \mathbb{C}\) and \(U_2 \otimes \mathbb{C}\) are isomorphic in a natural manner.
The above theorem implies that \(\text{Spp}_{\text{mod-} 2} (f)\) determines and is determined by the isomorphism class of the linear form \(L\) on \(U\). In order to verify the theorem the author considers the quadruplet \((U \otimes \mathbb{C}, b\otimes 1_\mathbb{C}, h\otimes 1_\mathbb{C},\text{Var} \otimes 1_\mathbb{C})\) instead of the Seifert form \(L\). This quadruple is the model of the generalized concept called hermitian variation structures. The theory of hermitian variation structures is developed in this article.
Reviewer: T.Urabe (Tokyo)

MSC:
14J17 Singularities of surfaces or higher-dimensional varieties
32S55 Milnor fibration; relations with knot theory
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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